The threefold numerical-effectivity conjecture for nef divisors

Let $X$ be a normal projective threefold with terminal singularities such that $K_X\sim0$. Let $L$ be a nef Cartier divisor on $X$ such that either $\nu(X,L)=1$, or $\nu(X,L)=2$ and $L\cdot c_2(X)=0$.

Threefold numerical-effectivity conjecture. Then $L$ is numerically equivalent to an effective divisor.

The paper identifies this as a further reduction of the Semiampleness Conjecture on threefolds. It remains open in the stated generality.